# Fields Medal Vladimir Voevodsky dies at age 51 in the USA

*Text from Institute for Advanced Study

Vladimir Voevodsky, a truly extraordinary and original mathematician who made remarkable advances in algebraic geometry, and whose most recent work concerned rewriting the foundations of mathematics to make them suitable for computer proof verification, died at age 51 on September 30 in Princeton, New Jersey. Voevodsky was Professor in the School of Mathematics at the Institute for Advanced Study, a position he held since 2002.

Voevodsky was able to handle highly abstract ideas to solve concrete mathematical problems. He had a deep understanding of classical homotopy theory, where the objects considered are flexible, meaning continuous deformations are neglected, and was able to transpose its methods in the very rigid world of algebraic geometry. This enabled him to construct new cohomology theories for algebraic varieties, which he used to prove the Milnor and Bloch-Kato conjectures, relating K-theory groups of fields and Galois cohomology.

“When I first saw the basic definitions in motivic cohomology I thought, ‘This is much too naïve to possibly work,’” said Pierre Deligne, Professor Emeritus in the School of Mathematics. “I was wrong, and Voevodsky, starting from those ‘naïve’ ideas, has given us extremely powerful tools.”

More recently, Voevodsky had worked in type-theoretic formalizations of mathematics and automated proof verification. He was working on new foundations of mathematics based on homotopy-theoretic semantics of Martin-Löf type theories. This led him to introduce a new, very interesting “univalence” axiom.

“Vladimir was a beloved colleague whose contributions to mathematics have challenged and enriched the field in deep and lasting ways,” said Robbert Dijkgraaf, IAS Director and Leon Levy Professor. “He fearlessly attacked the most abstract and difficult problems with an approach that was exceptionally innovative yet decidedly practical. Most recently, he was focused on developing tools for mathematicians working in highly advanced areas, such as higher-dimensional structures, laying out a grand vision for the future of mathematics. He was a pioneer and a catalyst and will be greatly missed by the Institute community.”

“Vladimir had the courage to think about the hardest and most fundamental problems in mathematics,” said Richard Taylor, Robert and Luisa Fernholz Professor in the School of Mathematics. “He would look for the right conceptualization, in the belief that then the difficulties would become surmountable. Very few mathematicians have succeeded in pulling off such an approach, but Vladimir succeeded magnificently, particularly in his construction of the derived category of mixed motives and his use of it to prove the Milnor and Bloch-Kato conjectures in K-theory.”

Born in Moscow on June 4, 1966, Voevodsky was awarded the Fields Medal in 2002 at age thirty-six, shortly after his appointment as Professor in the School of Mathematics. He had spent the prior three years (1998–2001) as a long-term Member.

Voevodsky earned the Fields Medal for developing new cohomology theories for algebraic varieties, which have provided new insights into number theory and algebraic geometry. His achievement was deeply rooted in the work of Alexandre Grothendieck, who in the 1960s revolutionized algebraic geometry. Grothendieck’s understanding of the notions of “space” and “localization” enabled him to construct for algebraic varieties over any field––where “continuity” does not make sense––cohomology groups similar to those known for complex algebraic varieties––where “continuity” makes sense. He constructed in fact a plethora of such theories, and dreamed that they should all be derived from a motivic theory, explaining their parallelism. By partially realizing this dream, Voevodsky gave us the powerful tool of motivic cohomology.

“Vladimir Voevodsky is an amazing mathematician,” observed Christophe Soulé, now Centre Nationnal de la Recherche Scientifique (CNRS) Research Director at the Institut des Hautes Études Scientifiques, when Voevodsky received his Fields Medal. “The field is completely different after his work. He opened large new avenues and, to use the same word as [Gérard] Laumon, he is leading us closer to the world of motives that Grothendieck was dreaming about in the sixties.”

A copy of Grothendieck’s “Esquisse d’un Programme,” which had been circulating among mathematicians since its submission to the CNRS in January 1984, was given to Voevodsky, then a first-year undergraduate at Moscow University, by his first scientific adviser, George Shabat. Voevodsky proceeded to learn some French for the sole purpose of reading it.

In 1990, Voevodsky and Michael Kapranov authored “∞-Groupoids as a Model for a Homotopy Category” in which they claimed to provide a rigorous mathematical formulation and a proof for Grothendieck’s idea connecting two classes of mathematical objects: ∞-groupoids and homotopy types. They later tried to apply similar ideas to construct motivic cohomology.

Kapranov arranged for Voevodsky to attend Harvard University as a graduate student, where he earned his Ph.D. in 1992 under his adviser David Kazhdan. Together with Andrei Suslin and Eric Friedlander, Voevodsky developed an approach to motivic cohomology that relied on a paper, “Cohomological Theory of Presheaves with Transfers,” authored by Voevodsky while he was a Member at the Institute in 1992–93.

Consequences of Voevodsky’s work, and two of his most celebrated achievements, are the solutions of the Milnor and Bloch-Kato conjectures, which for three decades were the main outstanding problems in algebraic K-theory. These results have striking consequences in several areas, including Galois cohomology, quadratic forms, and the cohomology of complex algebraic varieties. Voevodsky’s work may have a large impact on mathematics in the future by allowing powerful machinery developed in topology to be used for investigating algebraic varieties.

Voevodsky spent time at Harvard University as a junior fellow in the Harvard Society of Fellows from 1993–96 and as a visiting scholar from 1996–97 and again later from 2006–08. He was also a visiting scholar at the Max-Planck Institute in Bonn, Germany, from 1996–97, and Associate Professor at Northwestern University from 1997–98.

Voevodsky returned to the Institute as a long-term Member in 1998 and, in 1999–2000, gave a series of twenty lectures at IAS covering the foundations of the theory of motivic cohomology that he had developed with Suslin and Friedlander, subsequently published by Princeton University Press as *Cycles, Transfers, and Motivic Homology Theories*.

During these lectures, Voevodsky identified a mistake in the proof of a key lemma in his paper. Around the same time, another mathematician claimed that the main result of Kapranov and Voevodsky’s “∞-groupoids” paper could not be true, a flaw that Voevodsky confirmed fifteen years later. Examples of mathematical errors in his work and the work of other mathematicians became a growing concern for Voevodsky, especially as he began working in a new area of research that he called 2-theories, which involved discovering new higher-dimensional structures that were not direct extensions of those in lower dimensions. “Who would ensure that I did not forget something and did not make a mistake, if even the mistakes in much more simple arguments take years to uncover?” asked Voevodsky in a public lecture he gave at the Institute on the origins and motivations of his work on univalent foundations.

Voevodsky determined that he needed to use computers to verify his abstract, logical, and mathematical constructions. The primary challenge, according to Voevodsky, was that the received foundations of mathematics (based on set theory) were far removed from the actual practice of mathematicians, so that proof verifications based on them would be useless.

Voevodsky, who discovered an application of homotopy theory to the type theory used in computer proofs, had been working since 2005 on the ideas that led to the discovery of univalent models and gave the first public presentation on this subject at Ludwig-Maximilians-Universität München in November 2009. While he constructed his models independently, advances in this direction started to appear as early as 1995 and are associated with Martin Hofmann, Thomas Streicher, Steve Awodey, and Michael Warren.

In 2012–13, Voevodsky organized a special year in the School that focused on univalent foundations of mathematics, which resulted in a group of two dozen mathematicians writing a six-hundred-page book in less than six months. Voevodsky said that the main goal of his most recent work was “to advance the mathematical theory of dependent type theories to the level where it can be used for rigorous study of the complex type theories that are in use today and of the even more complex ones that will appear in the future.” Dependent type theories appear mostly in computer programs that use such theories as their foundation. Voevodsky formalized the mathematics in his papers using the proof assistant Coq and the UniMath library, which contains formalized mathematics for Coq of which Voevodsky was a co-founder and primary developer.

In addition to the Fields Medal, Voevodsky’s many contributions in the field of mathematics have been recognized by numerous honors and awards. He received a Sloan Fellowship from 1996–98, Clay Prize Fellowships in 1999, 2000, 2001, and many National Science Foundation grants for his work. Voevodsky also was named an honorary professor of Wuhan University (2004) and received an honorary doctorate from University of Gothenburg (2016). He was a member of the European Academy of Sciences.

Voevodsky is survived by his former wife, Nadia Shalaby, their two daughters, Natalia Dalia Shalaby and Diana Yasmine Voevodsky, his aunt, Irina Voevodskaya*,* and extended family in Russia and around the world. A gathering to honor Voevodsky’s life and legacy will take place at the Institute on October 8. A funeral service will be held in Moscow on December 27, followed by a mathematical conference in honor of his work on December 28 at the Steklov Mathematical Institute of the Russian Academy of Sciences. The Institute will convene an international conference on Voevodsky’s extraordinary and original work September 29–30, 2018.